Introductory Statistics Neil A. Weiss Ninth Edition

Introductory Statistics Neil A. Weiss Ninth Edition

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6.2 Areas Under the Standard Normal Curve 265 FIGURE 6.11 Finding the area under the standard normal curve that lies between 0.68 and 1.82 Area =? Area = 0.9656 0.2483 = 0.7173 = 0.68 = 1.82 = 0.68 = 1.82 Exercise 6.59 on page 268 Solution The area under the standard normal curve that lies between 0.68 and 1.82 equals the area to the left of 1.82 minus the area to the left of 0.68. Table II shows that these latter two areas are 0.9656 and 0.2483, respectively. So the area we seek is 0.9656 0.2483 = 0.7173, as shown in Fig 6.11. The discussion presented in Examples 6.3 6.5 is summaried by the three graphs in Fig. 6.12. FIGURE 6.12 Using Table II to find the area under the standard normal curve that lies to the left of a specified -score, to the right of a specified -score, and (c) between two specified -scores 1 2 Shaded area: Area to left of Shaded area: 1 (Area to left of ) (c) Shaded area: (Area to left of 2 ) (Area to left of 1 ) A Note Concerning Table II The first area given in Table II, 0.0000, is for = 3.90. This entry does not mean that the area under the standard normal curve that lies to the left of 3.90 is exactly 0, but only that it is 0 to four decimal places (the area is 0.0000481 to seven decimal places). Indeed, because the standard normal curve extends indefinitely to the left without ever touching the axis, the area to the left of any -score is greater than 0. Similarly, the last area given in Table II, 1.0000, is for = 3.90. This entry does not mean that the area under the standard normal curve that lies to the left of 3.90 is exactly 1, but only that it is 1 to four decimal places (the area is 0.9999519 to seven decimal places). Indeed, the area to the left of any -score is less than 1. Finding the -Score for a Specified Area So far, we have used Table II to find areas. Now we show how to use Table II to find the -score(s) corresponding to a specified area under the standard normal curve. EXAMPLE 6.6 Finding the -Score Having a Specified Area to Its Left Determine the -score having an area of 0.04 to its left under the standard normal curve, as shown in Fig. 6.13. FIGURE 6.13 Finding the -score having an area of 0.04 to its left Area = 0.04 Area = 0.04 =? = 1.75 265

266 CHAPTER 6 The Normal Distribution Solution Use Table II, a portion of which is given in Table 6.2. TABLE 6.2 Areas under the standard normal curve Second decimal place in 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.0233 0.0239 0.0244 0.0250 0.0256 0.0262 0.0268 0.0274 0.0281 0.0287 1.9 0.0294 0.0301 0.0307 0.0314 0.0322 0.0329 0.0336 0.0344 0.0351 0.0359 1.8 0.0367 0.0375 0.0384 0.0392 0.0401 0.0409 0.0418 0.0427 0.0436 0.0446 1.7 0.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0516 0.0526 0.0537 0.0548 1.6 0.0559 0.0571 0.0582 0.0594 0.0606 0.0618 0.0630 0.0643 0.0655 0.0668 1.5 Exercise 6.69 on page 269 Search the body of the table for the area 0.04. There is no such area in the table, so use the area closest to 0.04, which is 0.0401. The -score corresponding to that area is 1.75. Thus the -score having area 0.04 to its left under the standard normal curve is roughly 1.75, as shown in Fig. 6.13. The previous example shows that, when no area entry in Table II equals the one desired, we take the -score corresponding to the closest area entry as an approximation of the required -score. Two other cases are possible. If an area entry in Table II equals the one desired, we of course use its corresponding -score. If two area entries are equally closest to the one desired, we take the mean of the two corresponding -scores as an approximation of the required -score. Both of these cases are illustrated in the next example. Finding the -score that has a specified area to its right is often necessary. We have to make this determination so frequently that we use a special notation, α. DEFINITION 6.3 FIGURE 6.14 The α notation The α Notation The symbol α is used to denote the -score that has an area of α (alpha) to its right under the standard normal curve, as illustrated in Fig. 6.14. Read α as sub α ormoresimplyas α. Area = 0 In the following two examples, we illustrate the α notation in a couple of different ways. EXAMPLE 6.7 Finding α Use Table II to find a. 0.025. b. 0.05. Solution a. 0.025 is the -score that has an area of 0.025 to its right under the standard normal curve, as shown in Fig. 6.15. Because the area to its right is 0.025, the area to its left is 1 0.025 = 0.975, as shown in Fig. 6.15. Table II contains an entry for the area 0.975; its corresponding -score is 1.96. Thus, 0.025 = 1.96, as shown in Fig. 6.15. 266

6.2 Areas Under the Standard Normal Curve 267 FIGURE 6.15 Finding 0.025 Area = 0.975 Area = 0.025 Area = 0.025 3 2 1 0 1 3 3 2 1 0 1 3 0.025 =? 0.025 = 1.96 b. 0.05 is the -score that has an area of 0.05 to its right under the standard normal curve, as shown in Fig. 6.16. Because the area to its right is 0.05, the area to its left is 1 0.05 = 0.95, as shown in Fig. 6.16. Table II does not contain an entry for the area 0.95 and has two area entries equally closest to 0.95 namely, 0.9495 and 0.9505. The -scores corresponding to those two areas are 1.64 and 1.65, respectively. So our approximation of 0.05 is the mean of 1.64 and 1.65; that is, 0.05 = 1.645, as shown in Fig. 6.16. FIGURE 6.16 Finding 0.05 Area = 0.95 Area = 0.05 Area = 0.05 0.05 =? 0.05 = 1.645 Exercise 6.75 on page 269 The next example shows how to find the two -scores that divide the area under the standard normal curve into three specified areas. EXAMPLE 6.8 Finding the -Scores for a Specified Area Find the two -scores that divide the area under the standard normal curve into a middle 0.95 area and two outside 0.025 areas, as shown in Fig. 6.17. FIGURE 6.17 Finding the two -scores that divide the area under the standard normal curve into a middle 0.95 area and two outside 0.025 areas 0.025 0.95 0.025 0.025 0.95 0.025 0 0 =? =? 1.96 1.96 Exercise 6.77 on page 269 Solution The area of the shaded region on the right in Fig. 6.17 is 0.025. In Example 6.7, we found that the corresponding -score, 0.025, is 1.96. Because the standard normal curve is symmetric about 0, the -score on the left is 1.96. Therefore the two required -scores are ±1.96, as shown in Fig. 6.17. Note: We could also solve the previous example by first using Table II to find the -score on the left in Fig. 6.17, which is 1.96, and then applying the symmetry property to obtain the -score on the right, which is 1.96. Can you think of a third way to solve the problem? 267

268 CHAPTER 6 The Normal Distribution Exercises 6.2 Understanding the Concepts and Skills 6.45 Explain why being able to obtain areas under the standard normal curve is important. 6.46 With which normal distribution is the standard normal curve associated? 6.47 Without consulting Table II, explain why the area under the standard normal curve that lies to the right of 0 is 0.5. 6.48 According to Table II, the area under the standard normal curve that lies to the left of 2.08 is 0.0188. Without further consulting Table II, determine the area under the standard normal curve that lies to the right of 2.08. Explain your reasoning. 6.49 According to Table II, the area under the standard normal curve that lies to the left of 0.43 is 0.6664. Without further consulting Table II, determine the area under the standard normal curve that lies to the right of 0.43. Explain your reasoning. 6.59 Determine the area under the standard normal curve that lies between a. 2.18 and 1.44. b. 2 and 1.5. c. 0.59 and 1.51. d. 1.1 and 4.2. 6.60 Determine the area under the standard normal curve that lies between a. 0.88 and 2.24. b. 2.5 and 2. c. 1.48 and 2.72. d. 5.1 and 1. 6.61 Find the area under the standard normal curve that lies a. either to the left of 2.12 or to the right of 1.67. b. either to the left of 0.63 or to the right of 1.54. 6.62 Find the area under the standard normal curve that lies a. either to the left of 1 or to the right of 2. b. either to the left of 2.51 or to the right of 1. 6.63 Use Table II to obtain each shaded area under the standard normal curve. 6.50 According to Table II, the area under the standard normal curve that lies to the left of 1.96 is 0.975. Without further consulting Table II, determine the area under the standard normal curve that lies to the left of 1.96. Explain your reasoning. a. b. 6.51 Property 4 of Key Fact 6.5 states that most of the area under the standard normal curve lies between 3 and 3. Use Table II to determine precisely the percentage of the area under the standard normal curve that lies between 3 and 3. c. 1.28 1.28 d. 1.64 1.64 6.52 Why is the standard normal curve sometimes referred to as the -curve? 6.53 Explain how Table II is used to determine the area under the standard normal curve that lies a. to the left of a specified -score. b. to the right of a specified -score. c. between two specified -scores. 6.54 The area under the standard normal curve that lies to the left of a -score is always strictly between and. 1.96 1.96 2.33 2.33 6.64 Use Table II to obtain each shaded area under the standard normal curve. a. b. Use Table II to obtain the areas under the standard normal curve required in Exercises 6.55 6.62. Sketch a standard normal curve and shade the area of interest in each problem. 6.55 Determine the area under the standard normal curve that lies to the left of a. 2.24. b. 1.56. c. 0. d. 4. c. 1.96 1.96 2.33 d. 2.33 6.56 Determine the area under the standard normal curve that lies to the left of a. 0.87. b. 3.56. c. 5.12. 6.57 Find the area under the standard normal curve that lies to the right of a. 1.07. b. 0.6. c. 0. d. 4.2. 6.58 Find the area under the standard normal curve that lies to the right of a. 2.02. b. 0.56. c. 4. 1.28 1.28 1.64 1.64 6.65 In each part, find the area under the standard normal curve that lies between the specified -scores, sketch a standard normal curve, and shade the area of interest. a. 1 and1 b. 2 and2 c. 3 and3 6.66 The total area under the following standard normal curve is divided into eight regions. 268

6.3 Working with Normally Distributed Variables 269 a. Determine the area of each region. b. Complete the following table. Percentage of Region Area total area to 3 0.0013 0.13 3 to 2 2 to 1 1 to 0 0to 1 0.3413 34.13 1to 2 2to 3 3to 6.70 Obtain the -score that has area 0.80 to its left under the standard normal curve. 6.71 Obtain the -score that has an area of 0.95 to its right. 6.72 Obtain the -score that has area 0.70 to its right. 6.73 Determine 0.33. 6.74 Determine 0.015. 6.75 Find the following -scores. a. 0.03 b. 0.005 6.76 Obtain the following -scores. a. 0.20 b. 0.06 6.77 Determine the two -scores that divide the area under the standard normal curve into a middle 0.90 area and two outside 0.05 areas. 6.78 Determine the two -scores that divide the area under the standard normal curve into a middle 0.99 area and two outside 0.005 areas. 6.79 Complete the following table. 0.10 0.05 0.025 0.01 0.005 1.28 1.0000 100.00 In Exercises 6.67 6.78, use Table II to obtain the required -scores. Illustrate your work with graphs. 6.67 Obtain the -score for which the area under the standard normal curve to its left is 0.025. 6.68 Determine the -score for which the area under the standard normal curve to its left is 0.01. 6.69 Find the -score that has an area of 0.75 to its left under the standard normal curve. Extending the Concepts and Skills 6.80 In this section, we mentioned that the total area under any curve representing the distribution of a variable equals 1. Explain why. 6.81 Let 0 <α<1. Determine the a. -score having an area of α to its right in terms of α. b. -score having an area of α to its left in terms of α. c. two -scores that divide the area under the curve into a middle 1 α area and two outside areas of α/2. d. Draw graphs to illustrate your results in parts (c). 6.3 Working with Normally Distributed Variables You now know how to find the percentage of all possible observations of a normally distributed variable that lie within any specified range: First express the range in terms of -scores, and then determine the corresponding area under the standard normal curve. More formally, use Procedure 6.1. PROCEDURE 6.1 To Determine a Percentage or Probability for a Normally Distributed Variable Step 1 Sketch the normal curve associated with the variable. Step 2 Shade the region of interest and mark its delimiting x-value(s). Step 3 Find the -score(s) for the delimiting x-value(s) found in Step 2. Step 4 Use Table II to find the area under the standard normal curve delimited by the -score(s) found in Step 3. 269